Since there have been repeated claims that phasors are not vectors I decided to test this against the mathematical definition of a vector that has been in use for more than a century now.
According to my literature, this codification of vectors was first introduced by Peano in 1888 (Calcolo Geometrico) and strengthened by Weyl in 1918 (Space-Time-Matter).
This definition has been in use ever since.
A non-empty set of objects, T,
taken with a second non-empty set of objects F,
and two operations connecting these objects
is said to be a vector space and the objects of T are said to be vectors
if the following conditions (numbered 1 through 10) are satisfied by each and every object in T and F.
The objects of F are called scalars and are often (but not always) real or complex numbers.
One of the two operations is a rule which takes any two vectors u and v from set T and connects them to a third vector w, also from T.
We call this addition and use the + sign to denote it.
Thus w = u + v
The second operation connects vectors in the first set with scalars from the second and is called multiplication by a scalar. It takes any vector v from T and any scalar a from F and outputs the vector av, also in T.
I have used bold capitals for objects of V . the vectors
and normal, lower case for the objects of F .the scalars
The 10 axioms to be satisfied are
(1) if u and v are in T then w = (u+v) is in T
(2) u +v = v + u
(3) u + (v+w) = (u + v) + w
(4) There is an vector in T called the zero vector such that for all v in T
0 + v = v + 0 = 0
(5) For each and every vector, v in T there is a vector v such that
v + (-v) = 0
(6) If k is any scalar in F and v is any vector in T then there is a vector kv in T
(7) K(u + v) = ku = kv
(8) (k+l)v = kv + lv
(9) k(lv) = (kl)v
(10) There exists a scalar 1, in F, such that 1v = v
If I can find (define) a set of object obeying these rules I can call the object vectors.
Of course the rules are framed to include all the normal geometric and other sorts of vectors as subsets.
According to my literature, this codification of vectors was first introduced by Peano in 1888 (Calcolo Geometrico) and strengthened by Weyl in 1918 (Space-Time-Matter).
This definition has been in use ever since.
A non-empty set of objects, T,
taken with a second non-empty set of objects F,
and two operations connecting these objects
is said to be a vector space and the objects of T are said to be vectors
if the following conditions (numbered 1 through 10) are satisfied by each and every object in T and F.
The objects of F are called scalars and are often (but not always) real or complex numbers.
One of the two operations is a rule which takes any two vectors u and v from set T and connects them to a third vector w, also from T.
We call this addition and use the + sign to denote it.
Thus w = u + v
The second operation connects vectors in the first set with scalars from the second and is called multiplication by a scalar. It takes any vector v from T and any scalar a from F and outputs the vector av, also in T.
I have used bold capitals for objects of V . the vectors
and normal, lower case for the objects of F .the scalars
The 10 axioms to be satisfied are
(1) if u and v are in T then w = (u+v) is in T
(2) u +v = v + u
(3) u + (v+w) = (u + v) + w
(4) There is an vector in T called the zero vector such that for all v in T
0 + v = v + 0 = 0
(5) For each and every vector, v in T there is a vector v such that
v + (-v) = 0
(6) If k is any scalar in F and v is any vector in T then there is a vector kv in T
(7) K(u + v) = ku = kv
(8) (k+l)v = kv + lv
(9) k(lv) = (kl)v
(10) There exists a scalar 1, in F, such that 1v = v
If I can find (define) a set of object obeying these rules I can call the object vectors.
Of course the rules are framed to include all the normal geometric and other sorts of vectors as subsets.